Modélisation de la valeur client avec des approches ... · Modélisation de la valeur client avec...

Actes du 25e Congrès International de l’AFM – Londres, 14 et 15 mai 2009

Modélisation de la valeur client avec des approches déterministes et

stochastiques. Investigation de l'impact de l'hétérogénéité

Michel CALCIU

Docteur

IAE - Université des Sciences et Technologies de Lille - LEM

[email protected]

Francis SALERNO

Professeur

IAE - Université des Sciences et Technologies de Lille - LEM

[email protected].


Modélisation de la valeur client avec des approches déterministes et stochastiques.

Investigation de l'impact de l'hétérogénéité

Résumé (FR):

Ce papier fait un panorama des modélisations de la valeur actualisée du client et se concentre

sur sur deux principales méthodes de calcul: l'approche déterministe et stochastique. La

première ignore l'hétérogénéité des taux de rétention et/ou attrition au sein d'une cohorte. Elle

produit de formules faciles à utiliser par les managers et permet de résoudre une palette plus

large de problèmes manageriales. La deuxième approche apporte plus de précision dans les

calculs en intégrant l'hétérogénéité des taux de rétention et d'attrition, mais elle les mesure en

tant que variables actuarielles et non pas comme réponses à des efforts marketing. Nous

introduisons des formules pour calculer le nombre résiduel de transactions espérées

conditionné par le profil d'achat passé pour des modèles déterministes dans des situations

non-contractuelles et proposons des solutions pour évaluer les erreurs de prédiction par

rapport aux modèles stochastiques.

Mots clés : capital client, CRM, modélisation

Customer Litetime Value Modelling with deterministic and stochastic approaches.

Investigating the impact of ignored heterogeneity

Abstract (EN):

This paper gives a panorama of Customer Lifetime Value (CLV) modelling and focuses on

the two main categories of CLV calculation methods: the deterministic and the stochastic

approach. The first adopts simplified calculations that ignore heterogeneity in customers'

retention and/or churn rates within a cohort. It produces formulas that are easy to use by

managers and solves a larger number of managerial problems. The second approach brings

much more precision to CLV calculations by thoroughly considering retention and/or churn

rate heterogeneity, but measures them as actuarial variables and not as response to marketing

effort. We derive formulae to compute expected residual transactions conditional on customer

purchase profile for deterministic models in non-contractual contexts and suggest solutions to

evaluate forecasting error as compared to stochastic models.

Key words: .customer equity, CRM, modelling


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1.Introduction

The diffusion of interactive marketing and of data base technologies increased availability of

customer transaction data. This leads in most contexts to the adoption of customer and

customer portfolio evaluation methods that historically have emerged in the direct marketing

and catalogue sales industry. CLV is the discounted (actualised) value of cash flows generated

by a customer or a customer cohort during his «life» with a firm. The term Net Present Value

(NPV) has also been used in order to indicate that customer value is a financial measure

evaluating an investment (acquisition costs) that produces future cash flows. These cash flows

decline with time due to customers' churn. At the level of an individual customer or of a

cohort of customers CLV is usually calculated as a sum of discounted gains or cash flows

ignoring the initial investment or acquisition costs. Blattberg & Deighton (1996) coined the

term Customer Equity (CE) to designate the CLV from which acquisition costs are subtracted.

The same term has been used more recently in order to designate the net present value of the

customer base. As a company's customer base is formed dynamically by a sequence of

customer cohorts, in order to avoid confusions, Villanueva and Hanssens (2007) suggest using

the term Static Customer Equity (SCE) at individual customer of cohort level and the term

Dynamic Customer Equity (DCE) at the customer base level. This last measure as to Gupta et

al. (2004) is under certain circumstances a good proxy for the value of a company as it

accounts for both current and future relationships.In order to offer better definitions for the

CLV and CE concepts at individual customer (or cohort) level Pfeifer et al. (2005) introduce

the distinction between the value of a just-acquired customer and the value of a new or

existing one. If we note CLV as the value of a just-acquired customer (see table 1), that is a

customer value calculation starting just after the acquisition period, the value of a new or

existing customer includes also the margin (m) from the previous period (which for a new

customer is the acquisition period). CE subtracts acquisition costs from the value of a new

customer (these can be expressed as a fraction between prospect acquisition costs (A) and the

acquisition rate (a) that could be achieved due to these spendings).


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Tableau 1 – Customer value at various client/prospect states (adapted from Pfeifer & al.,2005)

Before presenting CLV calculations and modelling approaches, it is necessary to distinguish

between two types of temporal (dynamic) customer behaviour in his relationship with a

company. They have been named differently according to authors: "lost for good "/ "always

has share" by Jackson (1985), "retention model"/ "migration model " (Dwyer, 1989) or

"contractual"/"non-contractual" (Reinartz and Kumar, 2000).The retention model considers

that person or a company remains a customer as long as they generate transactions. The

migration model considers that customers can reappear (turn up again) after some periods

during which they did not make transactions and traces their probability of ‘reactivating’.

Our study is organised as follows. After a presentation of wider CLV modelling approaches,

two main categories of CLV calculation methods are analysed: the deterministic and the

stochastic approach. In the final part we develop formulae and apply them to investigate the

impact of ignoring heterogeity in non-contractual contexts using two datasets.

2.A review of CLV modelling approaches

Jain and Singh (2002), in one of the first and most frequently cited reviews of the CLV

literature, identify three main research directions. The first is the development of CLV

calculation models that focus on the revenue stream from customers, and on acquisition,

retention and other marketing costs in order to facilitate calculations, resource allocation and

optimisation of CLV. The second research stream, described as customer base analysis,

concentrates on methods that analytically predict the probabilistic value of customer

transactions from the existing customer base. The last direction uses analytical models in a

normative way and it analyses CLV implications for managerial decisions. As in the mean

time literature on the subject has largely increased and approaches have become more diverse,

other classification criteria and perspectives have been added. Gupta et al. (2006) for example

Measure of customer lifetime value Calculation

Value of a just-acquired customer CLV

Value of a new or existing customer m+CLV

Customer equity (CE) -A/a+m+CLV


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suggest a CLV analysis framework that takes as a starting point the customer acquisition, -

retention and - expansion (cross- and up selling) effects on customer value. Subsequently

interactions between these effects are analysed first at the individual and/or cohort level and

then at the firm level in order to determine the customer base value of the company. By

following this framework in order to structure modelling approaches the authors also suggest

a classification that distinguishes between classical RFM (Recency, Frequency and Monetary)

methods, stochastic- (probability), econometric-, persistence-, computer science – and growth

or diffusion models. By developing this same framework where customer acquisition,

retention and expansion through cross-selling and up-selling are seen as determinant of

Customer Equity, Villanueva and Hanssens (2007) classify CE models according to the kind

of data sources they deal with whether these are internal company databases, survey data,

company reports, panel data or even data on managerial judgments obtained by decision

calculus. The subarea of CLV modelling approaches that we are interested in may be called

CLV calculation models in the spirit of Jain and Singh (2002). As the diversity of CLV

calculation models is rather big, we adopt a progressive approach in dealing with these

models. We start with a simple generic model and increase complexity gradually by adding

dynamic customer behaviour contexts (contractual, non-contractual), transaction occasion’s

chronology aspects (discrete and continuous time), customer heterogeneity (deterministic and

stochastic models) as well as various analysis levels (individual, cohort or customer base). By

crossing CLV calculation models on two dimensions several modelling contexts can be

identified as shown in table 2.


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Industry Magazine subscriptions, Insurance

policies, etc.

Credit cards,

Mobile phone usage Contractual

Dynamic

Industry Catalogue sales, Events attendance,

Charity fund drives,

Retail purchases, Doctor visits,

Hotel stays

Type of

customer

relationship Non-

contractual Dynamic

Discrete Continuous

Transactions occasions

Source: adapted from Fader & Hardie (2008) p.10

Tableau 2 - Adequation between activities and/or industries and the typology of customer

relationships, examples.

The basic structural model (Jain et Singh, 2002) outlines the financial dimension of CLV:

CLV=∑t= 1

T M t− Ct

1+dt − 0 .5 (1)

where t = the customer transaction period; M = margin obtained and C = customer incumbent

costs in period t; n = the number of total periods (years) of the expected lifetime of a

customer.

The gain in each period is multiplied by a discount factor 1/(1+d) at a given power (here t-

0.5, which tries to fit the mean time between revenue and spending flows). This model is

general enough as the displayed periodic margin and cost, encapsulates various transactional

dynamics generating those financial flows. These flows can be the result of retention and

survival rates or probabilities when the context is contractual but can also result from

customer reactivation when the customer relationship context is non-contractual. As to Gupta

et al. (2006) CLV differs in two fundamental aspects from the purely financial net present

value measure. First it is estimated at an individual level or segment level and second it

explicitly integrates possible future customer attrition


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3.Deterministic and stochastic CLV models

The focus of the study is on deterministic and stochastic models belonging to the so-called

« buy till you die » framework (Schmittlein et al. ,1987; Fader & Hardie, 2008) which is

probably one of the most important research streams in CLV modelling nowadays. Both

assume individually constant buy and die probabilities for customers. While deterministic

models are suited for individual CLV calculations, stochastic models should be used when

computing CLV at an aggregated level (customer cohorts or customer base) as they integrate

heterogeneity of buy and/or die probabilities among individuals.

Deterministic retention models

More simple deterministic retention models are a good bases to introduce some fundamental

aspects of CLV calculations. By ignoring the temporal shift between retention gains and

spendings and regrouping them in order to form the net gain (g), that consists of the difference

between margin (m) and cost (c), and remains constant for a customer over time the CLV

calculation formula can become very concise. For a just acquired customer it is:

CLV=∑

t= 1

∞

gr t

1+dt=g

r1+d − r

(2)

where r is the retention rate and r/(1 + d − r) is called by Gupta and Lehman (2003) "margin

multiple". Gupta and Lehmann (2005) have shown that if gains increased with a constant rate

q the margin multiple becomes r/[1 + d– r(1 + q)]. By separating monetary flows from

transaction flows CLV can be expressed as the product of gains (here constant) and what has

been called by Calciu and Salerno (2002) discounted expected number of transactions or more

simply discounted expected transactions (DET). In other words CLV=g× DET . DET can be

seen as the CLV for transactions producing a one monetary unit gain (g = 1) . It can be used

as a proxy for CLV. DET for a just acquired customer is the margin multiple r/(1 + d − r) as

may be seen from formula (2). The DET for a new or existing customer is computed by

adding an initial transaction (see table 2). In order to predict future purchasing patterns by

those customers listed in the firm’s transaction database, and to generate estimates of their

expected lifetime value Fader, Hardie and Berger (2004) introduce the concept of « as-yet-to-

be-acquired » customers. A yet-to-be-acquired customer is seen as an existing customer that is

going to be retained (or acquired) which means that the DET of such a customer is the DET


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for an existing customer multiplied by the retention rate (see table 3).

Measure of discounted expected transactions (DET) Calculation

DET of a just-acquired customer DET(just acquired) = r/(1+d-r)

DET of a new or existing customer DET(new or existing) = 1+ DET(just acquired) = (1+d)/(1+d-

r)

DET of a yet-to-be-acquired customer (DERT) DET(yet-to-be-acquired)=r DET(new or existing) =

r(1+d)/(1+d-r)

Table 3 – Discounted Expected Transactions calculations retention models

The DET for a customer "yet-to-be-acquired " at a given moment n, just before the decision

whether to repurchase (or renew the contract), represents in fact the discounted expected

residual transactions (DERT) from that moment on. It allows to compute the Discounted

Expected Residual Lifetime value (DERL) (Fader & Hardie, 2007b) of existing customers for

which it is a proxy. DERT is well adapted for situations where the retention rates while

constant individually are heterogeneous within a cohort, as it can be shown that in such

situations aggregated retention rate increases over time. Deterministic retention models have

been extended by Gupta et al. (2004) in order to measure the value of the customer base of a

company or DCE (the Dynamic Customer Equity) , which is defined as the CLV of current

and future customers. DCE calculated as the sum of all cohorts taking into account new

customer acquisitions accounted for on an annual bases, can be obtained using the following

discrete time1 formula :

DCE=∑k= 0

∞ nk

1+dk∑t=k

∞

mt − kr t− k

1+dt − k–ck (3)

Where nk is the number of customers in cohort k. The initial number of customer in each

cohort is determined using a diffusion growth model.

While producing very useful managerial applications all determinist models as the one

1 The calculation for continuous time is:

DCE= ∫k= 0

∞

∫t=k

∞

nk e− dkmt− k e−1+d − r

rt− k

dtdk–∫k= 0

∞

nk e− dkck dk


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presented in this section when computing CLV at aggregated cohort or multi-cohort level

commit the error of ignoring heterogeneity of individual customer response probabilities. This

aspect is discussed later in this paper.

There is a fundamental marketing belief, that companies can influence customer response,

meaning that they can “control” customer acquisition and retention rates by varying

marketing efforts. This implies that it is also possible to optimally allocate marketing retention

and acquisition efforts. Formula 2 with retention rates expressed as a function of the retention

budget (r=f(R)) becomes:

CLV=m− R

r r1+d − r

=m− Rf R f R

1+d− f R (4)

The gain (g) is composed of the margin minus the average retention cost. It is expressed here

as the fraction of spendings per targeted customer (R) and achieved retention rate (r). This

formula helps find optimal retention spendings and is part of the elegant Blattberg and

Deighton (1996) model that also deals with optimal customer acquisition spendings using

formula 5.

CE = -A/a + m + CLV (5)

This model has recently been modified by Pfeifer (2005).

Deterministic migration models

The “always a share” behaviour is the alternate scheme to “lost for good” in what is known as

a dichotomy. Customer value comes here not only from surviving customers but also from

customers allowed to reactivate after a given number of inactive periods. Customers are

considered as “lost for good” only after exceeding that number of successive periods of

inactivity. By reducing the tolerated number of successive periods of inactivity to zero the

always a share model reduces to the lost for good model who can be seen as a special case of

this more general model. In non-contractual settings firms cannot know when a customer

becomes inactive. Intuitively they can apply rules (conventions) based upon recency,

frequency and monetary amount (RFM) of past purchases in order to decide whether a

customer is still active or not. By fixing recency, frequency and monetary states, based on past

behaviour, transition probabilities from one state to another can be computed and organised

into a matrix of transition probabilities in order to form a Markov Chain. If we use only


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purchase recency the migration scheme that controls the construction of the transition

probabilities matrix is shown in figure 1. It indicates that a customer in recency t can either

buy and pass in recency 1 or not buy and pass in an inferior recency state t+1.

R(1) <------R(t)-------->R(t+1)

Figure 1 - Migration scheme between Recency States

A detailed discussion of the matrix approach applied to customer migration can be found in

Pfeifer and Carraway (2000). One can also find there a detailed migration scheme between

R,F,M states and a graphical representation of transitions in and out of a given RFM state. A

practical application is given further in this paper. The matrix approach based on Markov

Chains in order to compute customer value has been used by Bitran and Mondschein (1996),

Pfeifer and Carraway (2000) using RFM variables in order to define transition states. Rust,

Lemon, and Zeithaml (2004) have defined transition probabilities matrix between brands that

vary over time following a logit model. Using the analogy between the retention probability

(r) and the transition probabilities matrix (P) it is easy to derive equivalent matrix formulas for

DET as the ones in table 4:

Measure of discounted expected transactions (DET) Calculation

DET of a just-acquired customer DET(just acquired) = [I − P /1+d− 1

− I ]

DET of a new or existing customer DET(new or existing) = I+ DET(just acquired) =

I − P/1+d− 1

DET of a yet-to-be-acquired customer (DERT) DET(yet-to-be-acquired)=P DET(new or existing) =

PI − P /1+d− 1

Table 4 – Discounted Expected Transactions calculations for migration models

While the formula for new and existing customers has been suggested by Pfeifer & Carraway

(2000), the other formulae are a contribution of this research. They will help compare

deterministic and stochastic approaches to CLV calculations, later in this paper.


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4. Stochastic CLV Models

Probability or stochastic models consider observed behaviour as the result of an underlying

stochastic process controlled by unobserved latent characteristics that vary across individuals.

As to Gupta et al. (2006), they try to find a simple paramorphic representation that describes

and predicts observed behaviour instead of trying to explain differences between explained

and observed behaviour as a function of covariables (as in regression models). They assume

that a given behaviour varies across the population according to a probability law. Their main

advantage is that they take into account and measure heterogeneity in observed behaviour.

Their main disadvantage is that by measuring the characteristics of those probability laws they

assume that it is only these laws that control observed behaviour and don't leave any place to

marketing variables as if the dynamic customer behaviour couldn't be changed by marketing

activities2. One of the most respected stochastic CLV models and a forerunner of recent

developments is the Pareto/NBD model that has been developed by Schmittlein, Morrison and

Colombo (1987). It is the benchmark model for non-contractual settings where transactions

can occur at any moment over time. It is not well suited for purchase situations that occur at

fixed time intervals and even less suited for contractual settings.

Contractual

Shifted Betageometric (sBGD-Fader

& Hardie, 2006,2007)

Beta-discrete-Weibull (BdWD - Fader

Expontial Gamma (EGD or Pareto

Distribution of second kind)

Type of

customer

relation-

ship Non-contractual

Betageometric Betabinomial

(BG/BBD -Fader, Hardie & Berger,

2004)

Pareto/NBD (Schmittlein, Morrison

& Colombo,1987),

Betageometric/NBD (Fader,Hardie,

2005),

Modified Betageometric/NBD

(Batislam et al. 2007)|

Discrete Continuous

Transaction cccasions

Table 5 - Stochastic Models covering various dynamic customer behaviour contexts

2 There are several studies that try to extend these models in order to integrate marketing covariates. Schweidel et al. (2008) use a mixed effects hazard model with both fixed and random components. Strictly speaking such models exceed the definition that is commonly given to stochastic models and should be classified as econometric models.


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The series of relatively recent developments undertaken by authors like Fader and Hardie, that

have begun with a modification of the Pareto/NBD the BG/NBD model (Fader, Hardie & Lee,

2005) that substantially simplifies parameter estimation, have lead these authors to

progressively cover all dynamic customer behaviour contexts presented in table 2 with

stochastic models matching each context and its corresponding industries . The names of

these models, their authors and the year of publication are given in table 5. If we take only the

prototype models (that are marked with bold characters) for each case in table 5 and try to

synthesize, we could say that they use as mixing distributions that account for the

heterogeneity in buy and/or die probabilities, beta distributions for discrete time models and

gamma distributions for continuous time models. For the simpler contractual case the

individual buying process representing the number of purchases during a given period for a

given buying probability3 follows a shifted geometric distribution in the discrete time case

and an exponential distribution in the continuous case. In the non-contractual settings the buy

and die probabilities are distinct and the probability distributions that represent the buying and

dying processes need to be integrated as subprocesses in order to represent the combined

behaviour. It follows that for the discrete case the individual buying process follows a

Bernoulli distribution and the dying process a geometric distribution while in the continuous

case the buying process is a Poisson distribution and the dying process is exponential. For

most of these models closed form expressions for the CLV could be derived as well as other

useful linked proxy measures like DET and DERT. For example the DERT for the stochastic

migration model in discrete time the beta-geometric/beta-binomial (BG/BB) distribution

model which will be used later in this paper in a comparison with the deterministic migration

model is given by the following formula:

Bα+x+ 1,β+n− x

Bα,βBγ,δ+n+ 1

Bγ,δ 2F11,δ+n+ 1,γ+δ+n+ 1,1/1+d/ Lα,β,γ,δx,n,m (6)

where (x,n,m) represent purchase history with x = number of purchases, m = recency, n= the

current period; (α,β,δ,γ ) are the estimated beta distribution parameters defining the

heterogeneous buying and dying probabilities; B(.) is the Beta function, 2 F1.is the

Gaussian hypergeometric function and L(.) is the Likelihood. Research on stochastic models

3 Which in contractual settings is equivalent to the (1- die) probability.


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for customer database analysis is very active. Existing models are modified or extended for

example Fader and Hardie (2007b) allow for the incorporation of time-invariant covariate

effects into Pareto/NBD and BG/NBD models. The BG/NBD model developed in 2005 as an

alternative to the Pareto/NBD model, has itself been modified and extended by several

researchers (Batislam et al. 2007, 2008; Fader and Hardie 2007b; Hoppe and Wagner 2007;

Wagner and Hoppe 2008). New ways to make parameters estimation and integration of

stochastic models are being explored. Several researchers use MCMC methods for this

purpose. For example Ma and Liu (2007) and Abe (2008) use the Gibbs sampler for parameter

estimation while Singh, Borle and Jain (2007) try to build a generalized framework for

estimating CLV when lifetimes are not observed using the Metropolis Hastings algorithm.

5.Investigating the impact of heterogeneity on CLV

In their yet unpublished paper called « Customer-Base Valuation in a Contractual Setting:

The Perils of Ignoring Heterogeneity » Fader and Hardie (2006) show that models that

assume constant retention at cohort or firm level are wrong as typically retention rates that can

be constant individually increase over time at the aggregate cohort level. They warn managers

and researchers not to use such models and suggest academics not to teach formulae for

computing CLV based on them. This is a rather strong point of view that seems to wipe out a

substantial area of research that has been dominant in the early stages of research in CLV

literature. We agree with all arguments showing that ignoring heterogeneity in customer

response rates is producing erroneous results and show how to extend exploring the precision

gap between those incriminated models and correct stochastic models in a non-contractual

setting. But by evaluating this precision gap we don't go as far as to ban those homogeneous

response models and calculation formulae and seek in this way a mean to keep them as we

think that they are more flexible, easier to use and understand and can integrate marketing

mix. In their above mentioned paper Fader and Hardie (2007) use the two datasets from a

previous version of their paper introducing the Shifted Beta Geometric distribution model

(Fader & Hardie, 2007) in order to compare the latter with the fixed retention rate model CLV

calculation given in formula 4. In order to extend the investigation of the impact of

heterogeneity to non-contractual situations we use the formula we have derived in table 3 for

computing Discounted Expected Residual Transactions (DERT) for a customer yet-to-be-


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acquired in order to produce Lifetime Value calculations for a customer dynamic behaviour

where buying and « death » probabilities are individually constant over time and “wrongly”

considered as homogenous within a cohort. Additional matrix formulations for expected

residual transactions at various discrete time periods are used in order to observe the

increasing precision gap between this model that ignores heterogeneity and the corresponding

stochastic model for non-contractual settings the "beta-geometric/beta-binomial " (BG/BB)

model (Fader, Hardie & Berger, 2004). In our investigation we also use two datasets. The first

is the Cruiseship dataset (Berger, Weinbert & Hanna,2003; Fader, Hardie & Berger, 2004)

consisting of cruise-line transactions for a cohort of 6094 customers over a period of five

years. The second is a catalogue sales dataset that for confidentiality reasons we will call

"Brabant". It consists of catalogue orders for a cohort of 36882 customers over seven seasons.

As opposed to the first dataset in the second one the buying behaviour shows high seasonal

effects with significantly more customers buying in autumn and less customers buying in

spring. As the BGBB model is not yet prepared to deal with such effects4 this opportunity is

used in order to point out some limitations that these stochastic models can have in dealing

with real world situations.

4 Extending the BG/BB model in order to integrate sesonality effects is a future reseach direction that we have started to explore.


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Source Fader, Hardie & Berger, 2004, p.10

(a) Observed buying behaviour (b) RF Transition Matrix (P)

Figure 2 Building the transition matrix from observed buying behaviour

The main contribution of this exercise is to adapt the homogeneous and constant transition

probabilities matrix approach to make it comparable to the corresponding stochastic approach.

We use the decision tree recording whether customers took cruises each year (Fig. 2a) to

compute the transition probabilities matrix (Fig. 2b). By applying expected residual

transactions calculations for both the deterministic migration model and the stochastic model

table 5 and figure 3 are obtained. From table 6 it can be seen that the model ignoring

heterogeneity undervalues the long term expected transactions and the discounted expected

transactions together with its proxy the CLV.


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Transactions

aft. 10 pds.

Expected

Transactions

DET Etats

determinist stochastic determinist stochastic determinist stochastic

r1f3 0.53 2.05 0.53 3.57 0.51 1.77

r1f2 0.69 1.52 0.69 2.65 0.62 1.31

r1f1 0.44 0.97 0.44 1.68 0.38 0.83

r2f2 0.55 1.28 0.55 2.23 0.53 1.10

r2f1 0.25 0.85 0.25 1.47 0.22 0.73

r3f1 0.10 0.72 0.10 1.25 0.09 0.62

r4 0.00 0.28 0.00 0.48 0 0.24

Table 6 - Expected transactions calculations for non-contractual settings

(a) determinist migration model (b) BG/BB Model

Figure 3 - Evolution of residual expected transactions using determinist and stochastic models

Figure 3 also shows that this undervaluation of the expected transactions is increasing with

the number of transaction periods. The same result is obtained for the catalogue sales dataset.

Models ignoring heterogeneity fail to recognise the sorting effect taking place in a

heterogeneous cohort that makes individuals with low buying probability and higher

probability to « die » leave the cohort earlier, so that over time the cohort progressively

retains customers with higher repeat buy probabilities and lower probabilities to die.

Therefore those models produce a downward-biased estimation of expected transactions and


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lifetime value.

The heterogeneity measures of stochastic models can also be used to evaluate the size of the

downward bias induced by deterministic models. In discrete time models the mixing

distribution used to account for heterogeneity is the beta distribution. It's two parameters α

and β can be characterized in terms of the mean µ = E(p) =α/(α + β) and the polarization index

φ = 1/(α + β + 1). The polarization index is a measure of heterogeneity. When α, β tend

towards zero the polarization index tends towards 1 and the beta distribution of the studied

probability p becomes U-shaped. Its values are concentrated near p = 0 and p = 1 meaning

high heterogeneity. Conversely when these parameters are big (α and β →∞) the polarisation

index tends towards zero and the beta distribution becomes a spike at its mean. In the case of

the Cruiseship dataset, beta distributions parameters for the purchase probability are α =

0.657, β = 5.193 indicating a mean buying probability of α/(α + β) =0.11 and a polarisation φ

= 1/(α + β + 1) = 0.146. The corresponding beta distribution parameters for the probability to

“die” γ = 173.761, δ = 1882.928 indicate that as regards the dying process this cohort is

highly homogeneous with a polarisation index near to zero and a spike at the mean probability

of 0.084. In the case of the catalogue sales dataset we have a higher mean buying probability

of 0.26 and a higher polarisation index 0.216 indicating more heterogeneity. The customer

death process is much less intense with a mean probability of 0.02 but with higher

polarization (0.18) than in the cruiseship case. This explains why long term expected

transactions and the DET measure for CLV are significantly higher for the catalogue sales

dataset. Further investigations of heterogeneity effects on long term expected transactions and

customer value can be made for each of the to dynamic processes that are combined in the

non-contractual settings, the buying process and the death process. A good illustration limited

only to the buying process is given in Fader and Hardie (2006).

6.Discussions and further research

The main purpose of this investigation of the impact of heterogeneity in non-contractual

setting was to show that it was possible to evaluate this impact and eventually how this could

be done. Obtaining a high degree of precision in such an exercise remains a future research

objective. While the parameter estimation methodology from purchase history data is well

defined for the stochastic migration model, this is not yet the case for the “deterministic”


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Markov chain model. From the few studies mentioned before that have applied Markov chain

transition matrix based migration models none seems to have suggested a particular

methodology for inferring transition probabilities from real data.

A potential solution to the problem of truncated transition probabilities would be to infer from

available data some causal relationship between purchase probabilities, recency and

frequency.

In general using deterministic CLV formulas on aggregated cohort data is error-prone and

should be avoided. Nevertheless there can be situations when a cohort's buying behaviour is

homogeneous. As segmentation seeking groups of customers with a homogeneous behaviour

is a fundamental marketing activity, there must be situations where cohorts are highly

homogeneous. Even in not particularly segmented data as the cruiseship dataset, the observed

cohort was highly homogeneous as concerns the death rate as indicated by the spike in the

estimated beta distribution. In such situations deterministic models could be used without

particular precautions. The use of stochastic models as precision benchmarks in CLV

calculation has been questioned in a recent study by Wübben & Wagenheim (2007). These

authors have compared predictive performance of stochastic non-contractual models

(Pareto/NBD and BG/NBD) to the one of simple managerial heuristics on three aspects: (1)

distinguish active customers from inactive customers , (2) generate transaction forecasts for

individual customers and determine future best customers, and(3) predict the purchase volume

of the entire customer base. They found that the simple heuristics perform at least as well as

the stochastic models with regard to all managerially relevant areas, except for predictions

regarding future purchases at the overall customer base level. The heuristics used were: (1)

the simple recency hiatus to distinguish between active and inactive customers; (2) the past

10% best customers in a customer base will also be the future 10%; (3) every customer

continues to buy at his or her past mean purchase frequency. Whilehaving better predicted

purchase volume at the overall customer base level (which is equivalent to customer equity)

as compared to the simple heuristics that extrapolates individual past purchases in the future it

is not sure whether stochastic models would have outperformed well tuned deterministic

models under the same conditions. This doubt is implicitly expressed by Wübben &

Wagenheim (2007) when they suggest that “it would be worthwhile to benchmark these

models against the approach that Gupta, Lehman, and Stuart (2004) use”. These deterministic


17

models provide flexible and easy to use formulas that could be substituted to simple

heuristics5 and under this cover coexist with the more sophisticated stochastic models. All this

justifies efforts towards making stochastic and deterministic CLV calculation models

comparable.

7.Conclusions

This study reviews CLV modelling approaches. It covers CLV computations at customer,

cohort and company customer base level. The managerial problems dealt with include

calculation aspects, CLV optimisation and optimal balance between customer acquisition and

retention spendings. During this review several concepts and formulae have been newly

organised and some new formulae have been derived. By introducing a clear distinction

between CE at individual and cohort level as opposed to the customer base level we help

disambiguate this term. The concept of residual expected transactions (Fader & Hardie, 2007)

after a given time period and for the long term is formalised for deterministic non-contractual

settings allowing for comparisons with corresponding deterministic and stochastic models.

The focus of the study is on deterministic and stochastic models belonging to the so-called

« buy till you die » framework (Schmittlein et al. ,1987; Fader & Hardie, 2008) which is an

active research stream in CLV modelling nowadays. Both assume in most cases individually

constant buy and die probabilities for customers. The deterministic models by ignoring

heterogeneity in buy and die probabilities that normally characterises cohorts of customers are

less sophisticated, easier to develop and implement. Consequently they offer solutions to a

larger area of managerial problems. An overview of these solutions is given in the paper in

order to outline problems that have not yet been solved with stochastic models. Deterministic

models are also easier to understand by managers as conceptually closer to simple heuristics.

While recognising the conceptual superiority and the quality of the measurement attained

through stochastic models we cannot oversee some difficulties in their estimation and at this

stage their limitations as concerns integration of marketing mix variables and sometimes their

lack of flexibility when adapting to real world situations. While some recent studies try to

5 A more detailed discussion of managerial customer base analysis heuristics and of the reasons why managers tend to use them instead of adopting more sophisticated academic methods can be found in Wübben & Wagenheim (2007).


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integrate stochastic models into more flexible frameworks like using MCMC methods for

parameter estimation, these models still need heavy development in order to achieve the

flexibility of deterministic models. This concern motivated us to find ways to explore the

impact of heterogeneity in non-contractual setting. Using the Markov Chain approach to

represent a deterministic customer migration model we derive formulae for computing the

residual expected transactions after a given time period and for the long term which in their

discounted form are a proxies for residual lifetime value calculations. In this way during each

(discrete time) period of the lifetime of a cohort the increasing precision gap between between

the deterministic migration model that ignores heterogeneity within a cohort and the

corresponding BG/BB stochastic model can be evaluated. Illustrations have been given for

two datasets. The observed deviations are rather high. This justifies baning the use of

deterministic models in CLV calculations at aggregated levels like cohorts or company

customer bases in non-contractual setting, as was suggested by Fader & Hardie who

investigated the impact of heterogeneity on CLV in contractual settings. Nevertheless we also

see in the possibility of evaluating the precision gap a potential mean to use deterministic

models as proxies for stochastic models in order to offer « satisficing » solutions to

managerial problems that cannot yet be efficiently solved with stochastic models.


19

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